Translations

The idea of a translation is a very common one in the practical world. Many machines are translational in their operation. The machinist who cranks the cutting-tool holder up and down the bed of the lathe, is "translating" it. The piston of an automobile engine is translated up and down in its cylinder. The chain of a bicycle is translated from one sprocket wheel to another as the cyclist pedals, and so on.

The bicycle chain is not only translated, it works because it has translational symmetry. After a translation of one link, it looks exactly as it did before. Because of this symmetry, it continues to fit over the teeth of the sprocket wheel (which itself has rotational symmetry) and to turn it.

Figure 1. Illustration by Hans & Cassidy. Courtesy of Gale Group.

Figure 2. Illustration by Hans & Cassidy. Courtesy of Gale Group.

Figure 3. Illustration by Hans & Cassidy. Courtesy of Gale Group.

One important use of translations is to simplify an equation which represents a set of points. The equation xy - 2x + 3y -13 = 0 can be written in factored form (x + 3)(y - 2) = 7. Then, letting x¹ = x + 3 and y¹ = y - 2, the equation is simply x¹y¹ = 7, which is a much simpler and more easily recognized form.

Such transformations are useful in drawing graphs where many points have to be plotted. The graph of x¹y¹ = 7 is a hyperbola whose branches lie entirely in the first and third quadrants with the axes as asymptotes. It is readily sketched. The graph of the original equation is also a hyperbola, but that fact may not be immediately apparent, and it will have points in all four quadrants. Many points may have to be plotted before the shape takes form.

If one has an equation of the form ax² + by² + cx + dy + e = 0 it is always possible to find a translation which will simplify it to an equation of the form ax² + by² + E = 0.

For example, the transformation x = x¹ - 2 and y = y¹ + 1 will transform x²+ 3y² + 4x - 6y - 2 = 0

into x² + 3y² - 9 = 0 which is recognizable as an ellipse with its center at the origin.

Transformations are particularly helpful in integrating functions such as intergral (x + 5)⁴ dx because intergral x⁴ dx is very easy to integrate, while the original is not. After the translated integral has been figured out, the result can be translated back, substituting x + 5 for x.

Translational symmetry is sometimes the result of the way in which things are made; it is sometimes the goal. Newspapers, coming off a web press, have translational symmetry because the press prints the same page over and over again. Picket fences have translational symmetry because they are made from pickets all cut in the same shape. Ornamental borders, however, have translational symmetry because such symmetry adds to their attractiveness. The gardener could as easily space the plants irregularly, or use random varieties, as to make the border symmetric, but a symmetric border is often viewed as esthetically pleasing.